Doubly stochastic measures with prescribed support

Seethoff, Terrance Lee; Shiflett, Ray C.

doi:10.1007/bf00533599

Cited by 24 publications

(20 citation statements)

References 6 publications

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“…(1 -A)x+ At~(x) evolve toward the second curve with A G [0,1]; compare with Seethoff and Shiflett's theorem [22]. In our case, the rule determining is that if dfi(x) ^ d^i (x) (as, e.g., for x £ S-2 in Fig.…”

Section: Introductionsupporting

confidence: 61%

Shape recognition via Wasserstein distance

Gangbo¹,

McCann²

2000

Quart. Appl. Math.

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Section: Introductionsupporting

confidence: 61%

Shape recognition via Wasserstein distance

Gangbo¹,

McCann²

2000

Quart. Appl. Math.

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“…This definition is equivalent to saying that there is no y 0 satisfying (12), i.e., the set of marked points in Let us conclude by recalling an example of an extremal doubly stochastic measure which does not lie on the graph of a single map, drawn from work of Gangbo and McCann [55] and Ahmad [3] on optimal transportation, and developed in an economic context by Chiappori, McCann, and Nesheim [27]. Other examples may be found in the work of Seethoff and Shiflett [92], Losert [69], Hestir and Williams [58], Gangbo and McCann [54], Uckelmann [101], McCann [76], and Plakhov [85].…”

Section: Uniqueness Of Optimal Transportationmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

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The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

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“…If the sum in (14) would also be zero (for all choices of ct\ β' eA ίn) ), then (a aβ -b aβ ) would have the properties (12), (13), (14), contradicting our minimal choice of (a aβ ) (since α αi « 2 -6 αχα2 = 0). Therefore φ aβ ) has the properties (12), (13), (14) (possibly with a smaller constant in (14)) and we will assume from now on that a aβ = b aβ for all a, β.…”

Section: Ua^mentioning

confidence: 88%

“…To exclude the second possibility, observe that Ui^o Dj contains with each element also the whole column corresponding to that element. Therefore, by (12):…”

Section: Ua^mentioning

confidence: 94%

“…It is along the lines of the results by Douglas in [5]. Then we study a special class of doubly stochastic measures, defined in a certain way by a measure preserving transformation T. A similar type of measures (but with a different behavior) has been studied in [12]. In Theorem 2 we give conditions on T that ensure that the corresponding measure be extremal (resp.…”

mentioning

confidence: 95%

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